Crystal field parameters and g-factors of the ground Kramers doublet of Сe3+ ion in LiYF4 crystal Текст научной статьи по специальности «Физика»

ISSN 2072-5981

Volume 17, Issue 1 Paper No 15103,

1-8 pages

2015

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Crystal field parameters and g-factors of the ground Kramers doublet

of Ce3+ ion in LiYF4 crystal

O.V. Solovyev

Kazan Federal University, Kremlevskaya, 18, Kazan 420008, Russia

E-mail: Oleg.Solovyev@mail.ru

(Received December 11, 2015; accepted December 19, 2015)

Analytical expressions are obtained for g-factors of the ground Kramers doublet of the impurity Ce3+ ions in LiYF4 crystal that rigorously take into account mixing of the 2F5/2 and F7/2 multiplets by crystal field. Dependence of g-factors on crystal field parameters is studied and possibilities of making conclusions about crystal filed parameters values on the basis of comparison with g-factors given in literature are considered.

PACS: 75.10.Dg, 76.30.-v

Keywords: crystal field parameters, g-factors, LiYF4:Ce3+

1. Introduction

Impurity Ce3+ ions substitute for Y3+ ions in LiYF4 crystal in sites with S4 point symmetry, in the nearest surrounding of the Y-site there are eight fluorine ions which form two deformed tetrahedrons. Spectrum of the Ce3+ ion 4f1 configuration consists of 7 Kramers doublets - F5/2 and F/2 multiplets splitted by S4 symmetry crystal field. Safe values of crystal field parameters for the Ce3+ ion in LiYF4 crystal are absent in literature due to difficulties in measuring of 4f crystal field energies for this compound. However, g-factors for the ground level of the Ce3+ ion in LiYF4 crystal were measured in [1, 2].

The goals of the present theoretical study are: 1) to derive analytical expressions for g-factors of the ground Kramers doublet of the Ce3+ ion in crystal field of tetragonal symmetry S4, rigorously taking into account mixing of the 2F5/2 and 2F7/2 multiplets by crystal field (unlike earlier papers, for example [2, 3], in which such a mixing was neglected and only the ground 2F5/2 multiplet was considered); 2) to investigate dependence of g-factors on crystal field parameters; 3) to consider possibility of making conclusions about 4f crystal filed parameters values for the LiYF4:Ce3+ crystal on the basis of comparison with the g-factors values for this compound measured in literature.

2. Analytical expressions for g-factors of the Ce3+ ion in crystal field of tetragonal symmetry $4

We consider an effective Hamiltonian H of the impurity Ce3+ ion consisting of spin-orbit interaction Hamiltonian

Hso =#SL, (1)

where £ is a 4f electron spin-orbit coupling constant, S and L are the spin and angular momentum of the 4f electron, and Hamiltonian of interaction with the S4 symmetry crystal field [4] (in crystallographic axes)

h cf = ^02c02) + B4C04) + B;C44) + B-4 C-4 + B6C06) + B46cf + B-4 C(? , (2)

where Bp are crystal field parameters, satisfying the equation Bp * = (-1)kBpk; C(kp) are components of one-electron spherical tensor operators

COO.

Note that there are only eight independent real quantities defining the effective Hamiltonian for the 4f electron in our model.

It is convenient to consider eigenfunctions | J, Jz) of the spin-orbit interaction Hamiltonian HSO as

a basis; here J and Jz are quantum numbers for the total momentum J of the 4f electron. Let us

express these eigenfunctions through the one-electron wavefunctions | m, a), where m and a are the

Crystal field parameters and g-factors of the ground Kramers doublet of Ce3+ ion in LiYF4 crystal magnetic and spin quantum numbers of the 4f electron:

5+ 5 2'"2

5

T 2

24--

(3)

5 7

Let us consider rigorously splitting of the multiplets J = — (ground multiplet) and J = — by

crystal field, taking into account mixing of states of these multiplets. There is no need to fulfill numerical diagonalization of the full matrix of the Ce3+ ion Hamiltonian H = H SO + H CF in the basis

| J, J^j. Looking ahead, we may state that it follows from calculations that the largest contribution to

3+ 5 5

wavefunctions of the ground 4f Kramers doublet of the Ce ion in LiYF4 comes from the -j, - ~

states (let us also note that the ground doublet corresponds to irreducible representations T5 and T6 of the double <S4 point group in Bethe notation; the first excited and second excited 4f levels have

symmetries (T7, T8) and (T5, T6) respectively). The state HCF (2) only with the states

5-,-2 ) is mixed by crystal field interaction

-7-,— ) and

2 2,

7 3'

—, - -j). Thus, it is sufficient to numerically

5 _ 3 2' 2/

diagonalize a matrix 4 x 4 of the Ce3+ ion Hamiltonian H = H SO + H CF (see (1) and (2)) with the following elements:

'5 5 i2'2

H

2,2--2Ç-2.Bo2 + 2.x, 2,-2

2 2 21 0 21 0 \2 2

H

2,-2 --2i+-Lm -7B

2 2 35 0 7 (

7 5

2,2

H

2,2 - lf.2L m; - 77 B«

2 2 2 21 0 77 0

25 429

B6

7 - -

- 2

H

7, - 3) - li-

1 3 15

R2 R4 ___ R6

—B--B--B ,

77

143

5 5

2,2

H

5 _ 1 2 - 2

Vl4 21

B44

5 5

2,2

H

7 5\ V6 B2 + 10^6 B4 5^6 B6

_/---B0 H--B0--B0,

2 2 21 0 231 0 429 0

5 5

2,2

H

7 3\ W35 „4 10V7

231

B44 -

143

m

5 - 3 ï - 2

H

7 5\ 8V2Ï 4 1^^/105

22

231

B-44 -

429

-B6

5 - 3 ? - 2

H

7 3\ V10 B2 + 8V10 B4 5V10 B6

B H--B--B,

22

35

231

143

7 5

2,2

H

7 3\ V2T0 B4 + 5/42 B6

22

77

429

(4)

Numerically diagonalizing the matrix with the elements (4) we obtain one of the wavefunctions of

the ground Kramers doublet of the Ce ion in LiYF4 in the form

I \ 1 r

M - N

5 5

-,-) + a 2 2

5 3 \ o -, — ) + ß

2 2'

7 5

r~2> + r

7 - 3 ^ - 2

, N -J 1 + \a\2 +ß|2

(5)

7

In (5) || < 1, ß\ < 1, H< 1 and, since the F7/2 multiplet is separated by the energy — £ from the

2F5/2 multiplet, it is expected that ß\ < ||, || < ||. The second wavefUnction of the ground Kramers

doublet of the Ce ion in LiYF4 can be obtained from (5) as

k)=\\v)=

5 _ 5 2' _ 2

a

5 3

2'2

7 _ 5 2' _ 2

■r

7 V> ^

2'2

(6)

5 5\ 5 3\ 7 ^ 7 3

2, 21' 2, 2/ , 2, 21, 2, 2

where Q is the time reversal operator [5]. The wavefunction can also be obtained directly by diagonalization of the Ce3+ ion Hamiltonian H matrix in the basis

with elements similar to (4) but with the opposite signs of crystal field parameters components and the

5 7

matrix elements cross between J = — and J = ^ states taken with the opposite signs themselves.

Now let us consider the Zeeman energy

HZe =Mb (2S + L)H, (7)

where H = (Hx,Hy,Hz) is magnetic field, /nB is the Bohr magneton. In the basis of the Ce3+ ion ground Kramers doublet states the Zeeman energy can be represented as

H Ze = A,g||Sf H +^(SfHx + SfHy), (8)

where Seff is the effective spin operator with S = 1/2, g and gL are g-factors when the magnetic

field is applied parallel and perpendicular to the tetragonal z-axis, respectively.

Successively calculating matrix elements {y1 |HZeand HZe(see (5), (6)) for the

Zeeman energy in the forms (7) and (8) we derived the following formulas for the g-factors of the ground Kramers doublet of the Ce3+ ion:

2

S =

7 N2

Sl =

7 N

(15_9|a|2 + 20|\ _ 12r (\ + \)_V1Ô(a\ +a>)), + V3Ôa\ + sÎ2r _ 16>/3\\.

(9) (10)

If we put P = y = 0 (9) and (10) transform into formulas obtained earlier [2, 3] for the case when mixing of the 2F5/2 and 2F7/2 multiplets is not considered. Such g-factors, calculated for the isolated 2F5/2 multiplet, depend only on || and satisfy the equation

1 i \2 1

— (g||- gl ) + 5g

lL g L '

(11)

16v " 7 5

where gL = 6/7 is the Lande g-factor for the ^5/2 multiplet. In a plane with axes ( S||, SL) (11) is an equation of an ellipse which will further be referred to as the «2F5/2 multiplet ellipse».

3. Investigation of dependence of g-factors of the Ce3+ ion in LiYF4 on crystal field parameters

In [1] the following values of g-factors for the ground Kramers doublet of the Ce3+ ion in LiYF4

crystal have been measured:

g^ = 2.737, sLxp = 1.475. (12)

Paper [2] gives close values g|Pp = 2.765, gLxp = 1.473. We will further rely on the [1] values (12). In the plane with axes (, gL) the experimental point (g^, gLxp ) lies inside the F5/2 multiplet ellipse (11) with the closest point of the ellipse ( = 2.779, Sl= 1.587) achieved for a = 0.5306.

Distance between the experimental point (12) and the closest point of the 2F5/2 multiplet ellipse amounts nearly 0.12, thus implying necessity of rigorous considering of mixing of the 2F5/2 and 2F7/2 multiplets and hence using the derived exact formulas (9), (10) for g-factors.

In [4] we used the following crystal field parameters in modeling of the interconfiguration 4f-5d absorption spectrum of the Ce3+ ion in LiYF4 (in cm-1):

B02 = 360 , B04 =-1400, B44 =-1240 + i-751, B06 =-67.2, B46 =-1095 + i-458. (13)

Signs and orders of magnitude of the parameters (13) are in agreement with parameters common in literature for trivalent rare earth ions doped in the LiYF4 crystal, obtained either from fitting experimental data (for example, [6]) or from microscopic calculations [7]. We will follow these crystal field parameters signs in our investigation, they are determined by geometry of the system in the first place. In [4] the value % = 625 cm1 was used for the spin-orbit coupling constant, this value also being in agreement with the common in literature (for example, 615 cm-1 in [6] and 628 cm-1 in [8]). Diagonalization of the matrix (4) with the Ce3+ ion Hamiltonian parameters used in [4] gives the following coefficients in the expansions (5), (6) of the ground Kramers doublet wavefunctions:

a = 0.4646 + i - 0.2787, | =-0.0304 - i - 0.0081, y = -0.0128 + i - 0.0061, (14)

and the following g-factors values, calculated according to (9), (10):

gn= 2.846, g±= 1.552. (15)

In accordance with small values of || and \y\ in (14), these g-factors lie very close to the 2F5/2 multiplet ellipse (11) - incidentally, as this was not the aim in [4] - with the closest point of the ellipse g = 2.849, g± = 1.560 achieved for || = 0.5148 being at distance of 0.009.

Note that signs of the real and imaginary parts of the a coefficient in wavefunctions (5), (6) of the ground Kramers doublet of the Ce3+ ion in LiYF4 should always be positive. Indeed, since the 2F7/2 multiplet is separated by the energy gap of 7% / 2 « 2200 cm-1 from the 2F5/2 multiplet, admixture of

5 - 3 ? - 2

(see (4)). Considering diagonalization procedure of the Hamiltonian matrix 2 x 2 on basis states

the

_state in wavefunction (5) is determined mainly by the matrix element y",'"

H

\ and

2 2

5

- —), it is easy to see that the real and imaginary parts of the coefficient a that

5 3\* VÏ4

B4* and, in accordance

2 2/ 21 4

defines this admixture have the same signs as 2H

with (13), are both positive.

Let us investigate dependence of g-factors calculated as (9), (10) on coefficients a, \ in wavefunctions (5), (6). The following question interests us: which coefficients values give g-factors that lie inside the 2F5/2 multiplet ellipse (11) at a distance of nearly 0.12, similarly to the experimental g-factors (12).

We use the realistic set of coefficients (14) established in [4] as a starting point and successively vary the real and imaginary parts of the coefficients; it is very convenient that corresponding g-factors (15) lie very close to the 2F5/2 multiplet ellipse and close to experimental values (12). Varying of Rea and Ima would give different points on the 2F5/2 multiplet ellipse (11), but not the shifts inside the ellipse that we are interested in, therefore we fix the a coefficient value at the level of (14).

In Fig. 1 g-factors calculated with the use of expressions (9), (10) are shown by triangles for Re \ ranging from _0.1804 to 0.4696 (66 values with the increment 0.01), as well as the experimental point (12) [1] and the F5/2 multiplet ellipse (11), which are also be shown in all figures below. The

starting point (15) of the variation procedure is highlighted by a round frame. As seen in Fig. 1, g-factors calculated for different Re ( values form

a distorted parabola in the plane (g , g±J with the

axis directed along the multiplet ellipse; the lower branch of the parabola corresponds to negative Re( values; shift by only « 0.05 inside the ellipse is reached at best. A very similar situation is found in Fig. 1 for g-factors shown by squares which correspond to Im ( ranging from

_0.3081 to 0.2919 (61 values with the increment 0.01). The calculated g-factors again form a distorted parabola in the plane (gy, g±j with the

axis directed along the 2F5/2 multiplet ellipse; the lower branch of the parabola corresponds to negative Im( values; shift by only « 0.04 inside the ellipse is reached at best.

A different situation is found in Fig. 2, where g-factors calculated as (9), (10) are shown by triangles for Ref ranging from _0.2128 to 0.1872 (41 values with the increment 0.01); the center point (15) of the variation procedure is highlighted by a round frame. As seen in Fig. 2, g-factors corresponding to different Ref values

form a distorted parabola in the plane (gy, g±j

with the axis directed across the 2F5/2 multiplet ellipse. Calculations show that the demanded shift by 0.12 inside the ellipse is reached at Re f «-0.17 (the lower branch of the parabola) or Ref«0.25 (the upper branch of the parabola). A very similar situation is found in Fig. 2 for g-factors shown by squares which correspond to Imf ranging from -0.1939 to 0.2061 (41 values with the increment 0.01). The

1.7-i

1.6-

tg

1.5-

1.4-

Experiment [1] Calculation [4]

F52 multiplet

—i—

2.6

—i—

2.8

3.0

'II

Figure 1. Diagram of g-factors of the ground Kramers doublet of Ce3+ ion in L1YF4: triangles - calculated for various Re (, squares - calculated for various Im (. In this Figure and below: line - g-factors for the isolated F5/2 multiplet, asterisk - experiment [1], round frame - calculation [4].

1.7

1.6

1.5

1.4

5/2 multiplet ellipse

2.6

2.8

3.0

'II

Figure 2. Diagram of g-factors of the ground Kramers doublet of Ce3+ ion in LiYF4: triangles - calculated for various Re y, squares - calculated for various Im y.

calculated g-factors again form a distorted parabola in the plane (, gwith the axis directed across the 2F5/2 multiplet ellipse. The demanded shift by 0.12 inside the ellipse is reached at Imy«-0.19 (the lower branch of the parabola) or Imy « 0.21 (the upper branch of the parabola).

The behavior of the g-factors in Fig. 1, Fig. 2 can be qualitatively understood from analysis of the expressions (9), (10). If we neglect the dependence of the normalization factor 1/ N (see (5)) on a, (, y coefficients, then gM is defined by a quadratic form of the ( coefficient for the fixed y coefficient (and visa versa), while gL is a linear form of the ( coefficient for the fixed y coefficient (and visa versa). Therefore we obtain a roughly quadratic dependence of gM on g± in Fig. 1, Fig. 2. The differences in the corresponding parabolas axes directions, which are nearly perpendicular for varying the ( and y coefficients separately, origin from different signs of (( and |y|2 in gM - see (9).

Comparison of Fig. 1 and Fig. 2 shows that it is the coefficient y in wavefunctions (5), (6) of the ground Kramers doublet of the Ce3+ ion in LiYF4 crystal that is foremost responsible for shifts inside the 2F5/2 multiplet ellipse towards the experimental g-factors (12). A simple suggestion can be made on the basis of the Ce3+ ion Hamiltonian matrix (4) analysis that large y values are achieved at large

values of the

5 5

2'2

H

matrix element in the first place, i.e. at large values of crystal field

1.7-i

"it Experiment [ 1] O Calculation [4]

parameters B44, Bf4. Of course, if both coefficients ( and y are sufficiently different from zero, non-linear effects that are more difficult to analyze occur, for example for Re(«-0.1 smaller absolute values of negative Rey or Imy coefficients are demanded (also «-0.1) to obtain the shift by 0.12 inside the ^5/2 multiplet ellipse.

Let us investigate numerically dependence of g-factors (9), (10) on crystal field parameters Bp (calculations show that dependence on the spin-orbit coupling constant £ is weak, besides the value of the latter is considered to be well established in literature - see for example [6, 8]). We use the set of crystal field parameters (13) given in [4] as a starting point (highlighted by a round frame in all figures) and successively vary all parameters yet keeping their signs, as has already been stressed above; the real and imaginary parts are varied independently for complex parameters with k ^ 0.

In Fig. 3 g-factors calculated for Re B^ ranging from -1995 cm-1 to -95 cm-1 (20 values with the increment 100 cm-1) are shown by triangles, larger absolute values of Re B46 correspond to smaller g values, and g-factors

1.6-

1.5-

1.4-

F multiplet

2.6

2.8 g

3.0

calculated for Im B® ranging from 8 cm 1 to 1958 cm-1 (20 values with the increment 100 cm-1) are shown by squares, larger values of Im B46 also correspond to smaller g values.

As seen in Fig. 3 the conclusion made above is confirmed that large absolute values of Re B® and Im B® lead to significant absolute values of the y coefficient and, consequently, significant shifts inside the ^5/2 multiplet ellipse on the g-factors diagram, towards the experimental g-factors (12) - compare with Fig. 2. As follows from Fig. 3 the necessary shift is obtained for Re B46 « -2000 cm1 or Im B46 « 2000 cm1.

A different situation is found for the B44 crystal field parameters. In Fig. 4 g-factors calculated for Re B44 ranging from -3640 cm-1

to -240 cm-1 (18 values with the increment 200 cm-1) are shown by triangles, larger absolute

Figure 3. Diagram of g-factors of the ground Kramers doublet of Ce3+ ion in LiYF4: triangles - calculated for various Re B46, squares - calculated for various Im B46.

2.0n

1.8-

1.6-

1.4-

1.2-

1.0-

0.8

F multiplet

5/2 r

Experiment [1 ] O Calculation [4]

1.2

—1—

1.6

2.0

—1—

2.4

2.8 3.2

—1—

3.6

g|

Figure 4. Diagram of g-factors of the ground Kramers doublet of Ce3+ ion in LiYF4: triangles - calculated for various Re B44, squares - calculated for various Im B44.

O.V. Solovyev

values of Re B4 correspond to smaller gM values, and g-factors calculated for Im B44 ranging from 351 cm-1 to 4751 cm-1 (23 values with the increment 200 cm-1) are shown by squares, larger values of ImB46 also correspond to smaller gM values. As seen in Fig. 4, unlike the case of B±4 parameters and

in contradiction with theoretical predictions given above, varying of B±4 parameters do not lead to expressed change of the y coefficient and g-factors do not demonstrate such behavior as in Fig. 2. For very large absolute values of Re B44 and Im B44 significant shifts inside the 2F5/2 multiplet ellipse on the g-factors diagram can, in principle, be obtained - see the left edge of the g-factors diagram in Fig. 4 -but such crystal field parameters values are impossible. The reason for discrepancy with our theoretical conclusions made above may lie in the fact that the B±4 crystal field parameters mix the

states

—, - — ) not only with the states 2 2

7 3 <

—,+2j, giving rise to the y coefficient in wavefunctions (5),

5 3

—,+—), and also mix the latter with the states 2 2

(6), but also with the states

2 2/

complicating the overall picture of mixing of the Ce3+ ion states by crystal field.

7

-,±-), thus 2 2

As follows from calculation, varying of crystal field parameters B^, Bq, B^ within reasonable

2

limits either leads to behavior of g-factors similar to that obtained in Fig. 1, with shifts along the F5/2 multiplet ellipse on the g-factors diagram, or simply does not affect significantly the calculated g-factors (this is the case for the B06 parameter).

Results obtained in this section for successive varying of different crystal field parameters were confirmed in a series of calculations with all crystal fields parameters being varied simultaneously (over 1 million sets of crystal field parameters values have been tested).

4. Conclusions

Analytical expressions have been derived for g-factors of the ground Kramers doublet of the Ce3+ ion in crystal field of tetragonal (S4) symmetry, rigorously taking into account mixing of the 2F5/2 and 2F//2 multiplets by crystal field: g-factors were expressed through coefficients of the ground doublet wavefunctions expansion in the eigenfunctions of the total momentum J of the 4f electron. Dependence of g-factors on the wavefunction coefficients values was investigated. These results can be applied to the entire class of compounds with Ce3+ (or Yb3+) centers in tetragonal crystal field.

Dependence of g-factors of the ground Kramers doublet of the Ce3+ ion in LiYF4 on crystal field parameters was studied, comparison with experimental data [1] was fulfilled. It was revealed that large values of the B®4 parameters are demanded to achieve agreement with the measurements results. However, such large values (ReB46 «-2000 cm-1, ImB^ « 2000 cm-1) do not seem reasonable. We suggest that to achieve agreement with experiment it is necessary to consider covalency effects for the impurity Ce3+ ion and introduce a reduction factor k for the orbital momentum of the 4f electron in the Zeeman energy [5] (compare with (7))

HZe = Mb (2S + kL ) H. (16)

Let us note, as an argument in favor of considering covalency effects in this compound, that the effective ionic radius of the Ce3+ ion is the largest for trivalent lanthanide ions, it amounts 1.143 A and is bigger than the ionic radius of the Y3+ ion (1.019 A) [9].

Another aspect of the problem of crystal field parameters determination for LiYF4:Ce3+ is that correct 4f energy levels for this compound should be provided. Experimental determination of the latter is a challenging task itself. Recently in [8] an attempt was made to determine several of the Ce3+ 4f crystal field levels from the measured with high spectral resolution 5d-4f luminescence spectrum in

the LiYF4:Ce3+ crystal at low temperatures: peaks observed in the luminescence spectrum were interpreted as zero-phonon lines corresponding to transitions from the lowest 5d state to different 4f levels of the Ce3+ ion. However, no modeling of vibrational structure of the luminescence spectrum was performed in [8], that could allow distinguishing zero-phonon and electron-vibrational transitions.

We are planning to fulfill a further investigation of 4f crystal field parameters in LiYF4:Ce3+ in the context of both g-factors values, which will be calculated taking into account covalency effects according to (16), and 4f crystal field levels, which will be improved over [8] by modeling the vibrational structure of the 5d-4f luminescence spectrum of this compound, in future.

Acknowledgements

The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University and the RFBR Grant 14-02-00826.

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